The number of integral terms in the expansion of $(5^{1/2} + 7^{1/6})^{642}$ is

If the coefficients of $r$-th and $(r+1)$-th terms in the expansion of $(1+x)^{24}$ are in the ratio $12:13$,then $r$ is the root of the quadratic equation

Let $\alpha > 0, \beta > 0$ be such that $\alpha^{3} + \beta^{2} = 4$. If the maximum value of the term independent of $x$ in the binomial expansion of $(\alpha x^{\frac{1}{9}} + \beta x^{-\frac{1}{6}})^{10}$ is $10k$,then $k$ is equal to

If the coefficients of $x^{-2}$ and $x^{-4}$ in the expansion of ${\left( {{x^{\frac{1}{3}}} + \frac{1}{{2{x^{\frac{1}{3}}}}}} \right)^{18}}, (x > 0),$ are $m$ and $n$ respectively,then $\frac{m}{n}$ is equal to

If $\frac{T_2}{T_3}$ in the expansion of $(a + b)^n$ and $\frac{T_3}{T_4}$ in the expansion of $(a + b)^{n + 3}$ are equal,then $n=$

If the second,third and fourth terms in the expansion of $(x+y)^{n}$ are $135$,$30$ and $\frac{10}{3}$,respectively,then $6(n^3+x^2+y)$ is equal to .............